About Confidence Interval for a Proportion Calculator
This calculator gives a confidence interval for a population proportion from the number of successes and the sample size, using the Wilson score or the Wald method. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
Use it for poll results, conversion rates, defect rates or any yes-or-no outcome. The page opens on 81 successes out of 263, a standard example from the literature, and shows Wilson and Wald side by side so you can see how they differ.
The Wald interval is p plus or minus z times the square root of p(1 − p)/n. It is simple, but it can be too narrow with small samples and proportions near 0 or 1. The Wilson interval fixes that, so it is the default here. A warning appears when fewer than 10 successes or failures make the Wald interval unreliable.
How to use Confidence Interval for a Proportion Calculator
- Choose what you want an interval for: a mean (standard deviation unknown or known), a proportion or a difference of means.
- Enter raw data or summary statistics. For a proportion enter the number of successes and the sample size.
- For a known-sigma interval, type the population standard deviation.
- Pick the confidence level: 90%, 95% or 99%.
- Read the interval and the margin of error, then open the working to see each formula.
Features
- Mean with unknown sigma (t interval) or known sigma (z interval).
- Proportion with the Wald and Wilson score methods, compared side by side, with a small-sample warning.
- Difference of two means, with the Welch method by default.
- Raw data or summary statistics.
- Margin of error, critical value, standard error and degrees of freedom shown.
- A bell curve with the central confidence region shaded.
- Step-by-step KaTeX working and a plain-English reading of the interval.
Wald and Wilson intervals
With p̂ = x / n and z the critical value:
Wald p̂ ± z · √(p̂(1 − p̂)/n)
Wilson [p̂ + z²/(2n) ± z · √(p̂(1 − p̂)/n + z²/(4n²))] / (1 + z²/n)Worked examples
81 successes in 263, 95% gives Wilson 0.2553 to 0.3662, Wald 0.2522 to 0.3638
Newcombe (1998), Statistics in Medicine 17:857, lists the Wilson interval 0.2553 to 0.3662.
520 successes in 1000, 95% gives Wilson 0.489 to 0.5508
3 successes in 20, 95% gives Wilson 0.05237 to 0.3604, Wald 0 to 0.3065
With so few successes the Wald interval is noticeably narrower and reaches closer to 0 than it should.
Privacy: how it works
- The interval is calculated by code in your own browser tab. Your numbers are not sent to Ulkaa or any other server.
- There is no account and nothing is stored once you close the tab.
Tips
- Use the t interval unless you truly know the population standard deviation. Knowing it is rare outside textbooks.
- For a proportion, prefer Wilson. Use Wald only when you are matching a textbook answer.
- If a difference interval contains 0, the data are consistent with no difference at that confidence level.
- Quadrupling the sample size roughly halves the margin of error.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your data are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your data are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
Which method is better, Wald or Wilson?
Wilson. Its actual coverage stays close to the stated level even for small samples, while the Wald interval is often too narrow. Use Wald only if a course or a report asks for it.
What if I have 0 successes?
The Wilson interval still gives a sensible upper limit. The Wald interval collapses to a single point at 0, which is clearly wrong.
More Confidence Interval Calculator pages
- Confidence Interval CalculatorGet a 90%, 95% or 99% confidence interval for a mean, a proportion or a difference of means, with every step shown.
- 95% Confidence Interval CalculatorType a mean, a standard deviation and a sample size and get the 95% interval and its margin of error.
- Confidence Interval for the Mean CalculatorPaste a list of values and get the confidence interval for the population mean.
- Confidence Interval for a Difference of MeansEnter two groups and get the interval for the difference between their population means.
- Z Confidence Interval CalculatorUse the normal distribution for a mean when you know the population standard deviation.